Operator Product States on Tensor Powers of $C^\ast$-Algebras

The program of matrix product states on tensor powers $\mathcal A^{\otimes \mathbb Z}$ of $C^\ast$-algebras, initiated in Comm. Math. Phys. {\bf 144}, 443-490 (1992), is re-assessed in a context where $\mathcal A$ is a generic nuclear $C^\ast$-algebra. For any shift invariant state $ω$, we demonstrate the existence of an order kernel ideal $\mathcal K_ω$, whose quotient action reduces and factorizes the initial data $(\mathcal A^{\otimes \mathbb Z}, ω)$ to the tuple $(\mathcal A,\mathcal B_ω= \mathcal A^{\otimes \mathbb N^\times}/\mathcal K_ω, \mathbb E_ω: \mathcal A \otimes \mathcal B_ω\to \mathcal B_ω, \bar ω: \mathcal B_ω\to \mathbb C)$, where $\mathcal B_ω$ is an operator system and $\mathbb E_ω$ and $\bar ω$ are unital and completely positive maps. Reciprocally, given a (input) tuple $(\mathcal A,\mathcal S,\mathbb E,ϕ)$ that shares similar attributes, we supply an algorithm that produces a shift-invariant state on $\mathcal A^{\otimes \mathbb Z}$. We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite site algebras, such as the group algebra of discrete amenable groups.